Semester Reflection
Content Skill
Over the course of this semester, we mastered many skills. One of the most challenging of these for me was the concept of doing math with infinity. I was able to grasp the fact that some infinites are bigger than others fairly quickly, however just throwing out anything that wasn’t multiplication, division, or exponential was an entirely new concept to me; and I didn’t like it. For example in a problem like the following, I would try to add all the smaller numbers to see which infinity was marginally bigger instead of just dropping them:
∞ + ∞2 + 33 – 2
∞ + ∞2 + 4444 – 33
At first I was convinced that these numbers still mattered, however, with Kyle’s coaxing, I eventually came to grips with the fact that infinity is so huge that any real number is nothing next to it and thus can have no effect on it if it isn’t applied in an exponent, through multiplication, or divides or is divided by infinity. I also had difficulty learning when such division problems become positive or negative infinity or zero. For example for the following problem, I might write zero instead of infinity because of this difficulty:
∞2
∞ + 2
One of Kyle’s metaphors finally hammered it home, “If you have infinite cookie and divide them amongst any number of friends you still have infinite cookies, but if you have one cookie to share with infinite friends, the cookie pieces eventually get so small, smaller that atoms or molecules, that it doesn’t matter and you all have zero cookies.”
Problem Solving Skill
Throughout the semester we also worked towards mastery of problem solving skills. One I still struggle with is recognizing and resolving errors. For me it is easy to drop a sign, or a number out of an equation, and this leads me to question how well I really know the content. For example, I made many of these types of mistakes while we were learning how to graph with derivatives, I made so many, in fact, that I never answered one of these questions completely correctly. Because of this I wasn’t aware that I actually knew what I was doing.
I tend to let little errors like forgetting about pluses or minus signs get in the way of my answers. This trend stems from the very methods I have always used in math. Since I first got to algebra, I’ve always just kept whatever answer I got to. I was always taught to go back and check my work, but I never did. I also tend to write very small and cram as much work as I can into as little space as possible. Both of these base methods propagate the problem.
However, I have begun to improve on this weakness by writing larger and more legibly. Because of this I am now less likely to misread my own handwriting and it is easier to go back and correct mistakes when I catch them. Yet the process of catching errors is still one I am unwilling to enter once I have gotten to an answer unless I am certain beyond all doubt that my answer is wrong. To further improve my skills at recognizing and resolving errors, this is a barrier I must and intend to hurdle.
Over the course of this semester, we mastered many skills. One of the most challenging of these for me was the concept of doing math with infinity. I was able to grasp the fact that some infinites are bigger than others fairly quickly, however just throwing out anything that wasn’t multiplication, division, or exponential was an entirely new concept to me; and I didn’t like it. For example in a problem like the following, I would try to add all the smaller numbers to see which infinity was marginally bigger instead of just dropping them:
∞ + ∞2 + 33 – 2
∞ + ∞2 + 4444 – 33
At first I was convinced that these numbers still mattered, however, with Kyle’s coaxing, I eventually came to grips with the fact that infinity is so huge that any real number is nothing next to it and thus can have no effect on it if it isn’t applied in an exponent, through multiplication, or divides or is divided by infinity. I also had difficulty learning when such division problems become positive or negative infinity or zero. For example for the following problem, I might write zero instead of infinity because of this difficulty:
∞2
∞ + 2
One of Kyle’s metaphors finally hammered it home, “If you have infinite cookie and divide them amongst any number of friends you still have infinite cookies, but if you have one cookie to share with infinite friends, the cookie pieces eventually get so small, smaller that atoms or molecules, that it doesn’t matter and you all have zero cookies.”
Problem Solving Skill
Throughout the semester we also worked towards mastery of problem solving skills. One I still struggle with is recognizing and resolving errors. For me it is easy to drop a sign, or a number out of an equation, and this leads me to question how well I really know the content. For example, I made many of these types of mistakes while we were learning how to graph with derivatives, I made so many, in fact, that I never answered one of these questions completely correctly. Because of this I wasn’t aware that I actually knew what I was doing.
I tend to let little errors like forgetting about pluses or minus signs get in the way of my answers. This trend stems from the very methods I have always used in math. Since I first got to algebra, I’ve always just kept whatever answer I got to. I was always taught to go back and check my work, but I never did. I also tend to write very small and cram as much work as I can into as little space as possible. Both of these base methods propagate the problem.
However, I have begun to improve on this weakness by writing larger and more legibly. Because of this I am now less likely to misread my own handwriting and it is easier to go back and correct mistakes when I catch them. Yet the process of catching errors is still one I am unwilling to enter once I have gotten to an answer unless I am certain beyond all doubt that my answer is wrong. To further improve my skills at recognizing and resolving errors, this is a barrier I must and intend to hurdle.
Problem of the Week
The problems of the week or POWs are difficult math problems that utilize concepts that we have yet to learn. They test our problem solving capabilities and our creative, mathematical minds. After we have come to an answer we them complete a write up in which we word the question asked in it's true mathematical form, write down our process with visuals, write down our answer and conclude wether or not said answer is correct. This POW is asks what the effects of a population control policy allowing people to have children until they have a son would be on said population. It assumed for the purposes of the problem that it is equally likely in all cases to have a son as it is to have a daughter, that couples did not split in any manner, and all had kids until they had a son.